Experimental Method

3.2.1 Fabrication and Characterization of Graphene Nanoribbons

Graphene monolayers were deposited on SiO2/Si (~290 nm/0.5 mm) substrates by mechani-cal exfoliation from natural graphite. Graphene thickness and GNR edge disorder were evaluated with Raman spectroscopy [45, 152, 153]. Samples were annealed in Ar/H₂ at 400 °C for 40 minutes. We used two approaches to define and fabricate graphene nanoribbon (GNR) arrays with pitch ~150 nm and varying widths: one with a PMMA mask (Fig. 3.1a), the other with an

Al mask (Fig. 3.1b) [154]. Double poly(methyl methacrylate) (PMMA) layers (PMMA 495K A2/PMMA 950K A4) were coated on the Si/SiO₂ substrate. For the electron (e)-beam lithogra-phy, we used 30 keV e-beam accelerating voltage. After opening 40 nm wide PMMA windows, we etched the graphene exposed through the windows with an oxygen plasma, creating GNRs of width W (Fig. 3.1a). This PMMA mask method was used for the W ≈ 130 nm, ~85 nm, and ~65 nm wide GNRs. For the narrower ~45 nm GNRs we used Al masks (Fig. 3.1b). In this case, after opening the PMMA windows, instead of plasma etching, we deposited 30 nm thick Al and obtained ~45 nm wide Al strips on graphene. After plasma etching of exposed graphene and Al etching (type A, Transene Company) we obtained ~45 nm wide GNRs. Figure 3.2 shows the atomic force microscopy (AFM) images of fabricated GNR arrays with W ≈ 130 nm, 85 nm, 65 nm, and 45 nm, respectively. The bottom and top regions of Figs. 3.2a and 3.2c correspond to the un-etched pristine graphene.

FIG. 3.1: Process to define the GNR widths. (a) PMMA mask method. (b) Al mask method.

Plasma etching

W

Al deposition and lift-off Plasma etching

Al etching W W

a

b

PMMA

graphene SiO2

Al

To characterize the prepared GNRs, we performed Raman spectroscopy with a 633 nm wavelength laser (~1 µm spot size) as shown in Fig. 3.3. Even before patterning into GNRs, we selected only monolayer graphene flakes, identifiable through their 2D (G’) to G Raman peak ratio, and through a single fitted Lorentzian to their 2D (G’) peak. The unpatterned graphene samples had no identifiable D peak, indicative of little or no disorder [155]. On the other hand, the GNR arrays showed a pronounced D band consistent with the presence of edge disorder [152]. The peak intensity of the D band with respect to that of the G band increases with narrow-er GNR width. Because the edges of graphene snarrow-erve as defects by breaking the translational symmetry of the lattice, the larger fraction of the edge in narrow GNRs will enhance the D peak [156]. The inset of Fig. 3.3a quantitatively shows the behavior by calculating the ratio of inte-grated D band (ID) to G band (IG), ID/IG, as a function of GNR width (symbols). The width FIG. 3.2: Atomic force microscopy (AFM) images of GNR arrays. (a) W ~ 130 nm GNRs. (b) W

~ 85 nm GNRs. (c) W ~ 65 nm GNRs. Inset: AFM image near metal electrodes. (d) W ~ 45 nm GNRs. The axis units are given in microns on each panel.

graphene a

d

0.0 0.2 0.4 0.6 0.8 1.0 1.2

c

0.0 0.2 0.4 0.6 0.8 1.0 1.2

b [µm]

dependence of the peak ratio follows a relation of ID/IG = cW^-1 with c = 210 nm (dashed line), which is consistent with previous reports of GNR characterization [95, 152]. Figure 3.3b shows the D, G, and D’ peaks in detail of fabricated GNR arrays and un-patterned graphene with 633 nm wavelength laser. Figures 3.3c-g show the Raman 2D band spectra (scattered points) for the samples. All 2D bands are fit by a single Lorentzian peak (solid red curves) with ~2650 cm^-1 peak position, which is consistent with previous reports of monolayer graphene and GNRs [152].

FIG. 3.3: Raman spectra of GNR arrays and un-patterned graphene. (a) Raman signal for W ~ 130, 85, 65, 45 nm GNRs and un-patterned graphene. Each spectrum is vertically offset for clarity. Inset is the ID/IG ratio as a function of GNR width, consistent with the enhanced role of edge disorder in narrower GNRs [105, 152, 153]. (b) Zoomed-in D, G, and D’ bands of all samples. (c-g) 2D bands with a single Lorentzian fit for all samples, consistent with the existence of monolayer graphene.

3.2.2 Thermometry Platform and Measurements

To fabricate thermometry platform for thermal measurements, electron (e)-beam lithography was used to pattern the heater and sensor thermometers [54, 157] as long, parallel, ~200-nm-wide electrodes with current and voltage probes, with a separation of L ≈ 260 nm. Electrodes were deposited by successive evaporation of SiO2 (20 nm) for electrical insulation and Ti/Au (30/20 nm) for temperature sensing. Figure 3.4 illustrates scanning electron microscopy (SEM) images of our several experimental test structures, showing graphene and GNR arrays supported

FIG. 3.4: Measurement of heat flow in graphene ribbons. (a) Scanning electron microscopy (SEM) image of parallel heater and sensor metal lines with ~260 nm separation, on top of graphene sample (colorized for emphasis). A thin SiO₂ layer under the metal lines provides electrical insulation and thermal contact with the graphene beneath. (b) Similar sample after graphene etch, serving as control measurement for heat flow through contacts and SiO2/Si underlayers. (c) Heater and sensor lines across array of graphene nanoribbons (GNRs). (d) Magnified portion of array with GNR widths ~65 nm; inset shows atomic force microscopy (AFM) image of GNRs. Scale bars of (a-d) are 2 μm, 1 μm, 2 μm and 1 μm, respectively. (e) Three-dimensional (3D) simulation of experimental structure, showing temperature distribution with current applied through heater line.

a

b

∆T (K)

c d

e

10 1 0.1

290 nm SiO₂ Silicon graphene

SiO₂

on a SiO2/Si substrate, as well as bare SiO2 with graphene etched off by an oxygen plasma (however, graphene still exists under the metal electrodes, consistent with the other samples).

Figure 3.5 shows the measurement set-up for the SiO2 sample (Fig. 3.4b) as an example. To block environmental noise including electrostatic discharge, π-filters with a cut-off frequency of 2 MHz were inserted across all measurement lines. To control the temperature, Physical Property Measurement System (PPMS, Quantum Design) was used with a temperature range of 10 K – 363 K. Inside the PPMS, the vacuum environment is always a few ~10^-3 Torr, rendering convec-tive heat losses negligible.

The measurement proceeds as follows. In the heater, we apply a sinusoidal voltage with fre-quency lower than 2 mHz through a standard resistor of 1 kΩ to flow current with a range of

±1.5 mA, generating a temperature gradient across the sample. To obtain the response of the FIG. 3.5: Scanning electron micrograph (SEM) of thermometry platform and measurement configuration. Scale bar is 4 µm. Image taken of sample after graphene was etched off, and after all electrical and thermal measurements were completed. Dark region around “part 1” is substrate charging due to previous SEM imaging performed to obtain Fig. 3.4b in the main text.

V_H

Lock-in amplifier

2.147 kHz

1 kΩ

< 2 mHz

π-filter sensor

heater

1 MΩ

sensor (Fig. 3.6a), we measured its resistance change by a standard lock-in method with excita-tion frequency 2.147 kHz and root-mean-square (rms) current 1 μA (carefully chosen to avoid additional heating). All electrical measurements were performed in a four-probe configuration.

Both electrode resistances are calibrated over the full temperature range for each sample (Fig.

3.6b), allowing us to convert measured changes of resistance into changes of sensor temperature ΔTS as a function of heater power PH (Fig. 3.6c). We sometimes found that the electrical re-sistance of the sensor slowly drifted (increased) with time at room temperature. However, this effect was stabilized after annealing the sample at 363 K for 5 min, eliminating resistance drift at room temperature. Therefore, this behavior could be related to the absorbed water on the metal electrodes.

3.2.3 Experimental Data and Error

Figure 3.6a shows the measured sensor resistance change, ∆R_S, as a function of the power applied to the heater, PH, at T = 100 K for the SiO2 sample (Fig. 3.4b). The black (for negative heater current, I_H) and red (for positive I_H) lines overlap with each other, indicating the meas-urement is symmetric and reliable. The calibration for sensor and heater resistance vs. tempera-ture is shown in Fig. 3.6b; thus, sensor heating due to heater power ∆RS can be converted to a temperature rise, ∆TS, as shown in Fig. 3.6c by using the resistance-temperature calibration curve. The fitted slope of the ∆TS vs. PH curve in Fig. 3.6c is 0.01797 K/µW, which is then used for the extraction of thermal properties through simulations (see Section 3.3). Figure 3.6d shows the measured ratio of PH to ∆TS for all representative samples as a function of ambient tempera-ture from 20 K to 300 K. The uncertainty of the electrical thermometry measurement is ~2%

(Tables 3.1 and 3.2), which is comparable to the symbol size on this plot. Thus, although data are available down to 20 K, the values are distinguishable without ambiguity only when T ≥ ~70 K,

which is the main temperature range shown in Section 3.5.

We note that P_H/ΔTS shown in Fig. 3.6d is not the thermal conductance through graphene, because ΔTS is the temperature rise in the sensor, not the temperature drop from the heater to sensor, and P_H is the heat generated in the heater, not the one flowing in graphene. The thermal FIG. 3.6: Measurement process. (a) Sensor resistance change, ∆RS as a function of heater power, PH at T =100 K for the SiO2 sample (Fig. 3.5). Red and black lines are taken with current flow in opposing direction. (b) Calibration of sensor and heater resistances as a function of temperature.

The inset shows the R-T curve and slope of the sensor near T = 100 K. (c) Converted sensor temperature rise, ∆TS as a function of heater power, PH at T = 100 K from (a) and (b). The slope of the fitted red line is ∆TS/P_H = 0.01797 K/µW, which is later used to extract the thermal properties of the SiO2 layer (see Fig. 3.9). (d) Measured ratio of heater power to sensor tempera-ture rise for all representative samples. The uncertainty of these data is ~2% (Tables 3.1 and 3.2), comparable to the symbol size. Although this plot shows all raw data taken, the values can be distinguished without ambiguity only at T ≥ 70 K, which is the temperature range displayed in the Figs. 3.12 and 3.16.

conductance of the graphene cannot be immediately extracted from our raw data, due to heat leakage into the substrate (a drawback of the substrate-supported thermometry method). Instead, we employ 3D simulations to carefully account for all heat flow paths and, by comparison with the experiments, to obtain the thermal conductance of the graphene samples (see Section 3.4).

Figure 3.7a shows the sensor resistance as a function of count number (time) without apply-ing current to the heater at T = 102 K. The standard deviation of the scattered data points is δR = 3.1 mΩ, which corresponds to δT ~ 36 mK by using calibration coefficient 0.0866 Ω/K obtained in Fig. 3.6b. Thus, the error of the temperature reading is ±36 mK, primarily due to slight ambi-ent temperature fluctuations in the PPMS (consistambi-ent with a fluctuation of ±30 mK of the dis-played temperature on the PPMS monitor). Zooming into the circled region of Fig. 3.7a, we note a resistance fluctuation δR = 0.17 mΩ, corresponding to a temperature uncertainty ±2 mK due to electrical measurement instruments. Therefore, during the time scales of most of our measure-ments our temperature accuracy is limited by the ambient temperature control of the PPMS rather than by the electrical measurements themselves.

FIG. 3.7: Measurement error and thermal steady-state. (a) Sensor resistance as a function of

The sweep speed of the heater power is chosen to be sufficiently slow to reach thermal steady-state between the heater and sensor. Figure 3.7b shows the heater power (P_H) sweep with time and corresponding resistance change in the sensor, ∆RS. Data shown here correspond to the linear ramp in Fig. 3.6a. After ~15 minutes, the heater power reaches its maximum, and the change of sensor resistance follows the same trend without delay, indicating that the thermal steady-state between the heater and the sensor is established during the entire sweep process. If the sweep speed of the heater power is too fast to reach the steady-state, the data point at P_H ~ 110 µW in Fig. 3.6a will deviate from the linear trend. We also verified this by a comparison between the corresponding constant DC power and the above methods.

3.3 Model to Analyze Experimental Data